The Obama administration has given the go-ahead for ten of America's biggest banks to repay $68bn (£43bn) in emergency bail-out money after judging that the institutions have recovered sufficient stability to survive without a financial crutch from taxpayers.

JP Morgan, Morgan Stanley and Bank of New York Mellon are among those handing back government money pumped into top Wall Street firms to avert financial collapse at the height of last year's meltdown in the banking industry.

The ability of the banks to return funds has been widely greeted as a sign that Wall Street is edging towards a recovery from its most traumatic financial crisis since the Great Depression of the 1930s. But the US treasury secretary, Timothy Geithner, echoed analysts and industry insiders by warning that the reconstruction of the banking sector is far from complete.

"These repayments are an encouraging sign of financial repair, but we still have work to do," said Geithner.

Those repaying funds have been obliged to meet stringent challenges. They were required to prove that they can raise money independently on the public markets and Treasury officials scrutinised the adequacy of their capital in financial "stress tests" carried out last month.

Many banks are anxious to hand back the money as fast as possible, keen to escape political oversight, restrictions on dividend payouts and a perceived stigma attached to reliance on taxpayers' funds.

"If I was a bank, I'd rather be out than in," said Jason Goldberg, a banking analyst at Barclays Capital in New York. He pointed out that banks have raised more than $50bn in private capital over recent months but he added that they still face risks arising from bad debt on credit cards and mortgages: "Talk about nationalisation and liquidity has abated but they still face the prospect of rising loan losses."

The Treasury did not name the banks repaying money amid concern that those left in the $700bn government support programme could be further tainted through the emergence of a two-tier financial industry. But many of those involved were quick to identify themselves.

JP Morgan, which has bought struggling rivals such as Bear Stearns and Washington Mutual, said it was returning the government's entire $25bn investment in its preferred shares. Chief executive Jamie Dimon said: "We feel it's best for our government to be able to use these funds for other critical purposes."

Morgan Stanley is handing back $10bn, Bank of New York Mellon is repaying $3bn, State Street is refunding $2bn and Northern Trust said it was returning $1.58bn. Goldman Sachs and American Express are thought to be among the others repaying money. Scores more banks, however, remain dependent on government funds – including Citigroup and Bank of America, which have been told to deepen their capital cushions significantly before repaying Tarp money.

The Treasury said it would use the refunded money to replenish its funds to respond to any future financial instability. Analysts caution that there is still a significant risk that further economic deterioration could undermine banks.

Nancy Bush, a banking analyst at NAB Research, said: "There are still forces which could come back to haunt us."

The White House is set to announce plans next week for longer term reforms of Wall Street including a shake-up of regulatory bodies and tighter oversight of executive bonus payouts, which have been blamed for exacerbating a culture of excessive risk-taking.

In testimony to the US senate's appropriations committee , Geithner promised a comprehensive plan to "ensure that no large and interconnected firm or market can take on so much risk that its failure could destabilise the entire financial system". The treasury is expected to appoint a "compensation czar" to police pay packages and the Securities and Exchange Commission is likely to get new powers to oversee remuneration.

The US stockmarket was unimpressed by the banks' repayment of ­government funds. By midday, the Dow Jones ­Industrial Average was down 33 points to 8,370.

The Congressional Oversight Panel expressed concern that the government's stress tests on banks may have been too lenient. It pointed out that the government tested banks' ability to withstand an unemployment rate of 8.9% – yet the rate has risen to 9.4%. The panel's chairman, Elisabeth Warren, told lawmakers: "The worst case scenario number for 2009 is, in fact, not the worst case. We're going to see worse numbers."

One view among experts is that stress tests should become a regular event. "The best thing that could happen is some permanent form of stress testing," said Robert Gach, head of global capital markets at the consultancy firm Accenture. "We're starting to see a corner turned but it's still going to be a bumpy road going forward."